arXiv:quant-ph/9608015 · Modern Physics Letters A

Instantonic Approach to Triple-Well Potential

Soo-Young Lee, Jae-Rok Kahng, Sahng-Kyoon Yoo, D.K. Park, C. H. Lee, Chang Soo Park, Eui-Soon Yim

Published Aug 10, 1997• 7 citations

Abstract

By using the usual instanton method, we obtain the energy splitting due to quantum tunneling through the triple-well barrier. It is shown that the term related to the midpoint of the energy splitting in propagator is quite different from that of double-well case, in that it is proportional to the algebraic average of the frequencies of the left and central wells. The result of the energy splitting is consistent with the known variational calculation.

DOI: 10.1142/s0217732397001837

Related papers

Instantonic approach to triple well potential

By using a usual instanton method we obtain the energy splitting due to quantum tunneling through the triple well barrier. It is shown that the term related to the midpoint of the energy splitting in propagator is quite different from that of double well case, in that it is proportional to the algebraic average of the frequencies of the left and central wells. Typeset using REVTEX 1 I.

Jan 1, 19965 citations

Tunneling and energy splitting in an asymmetric double-well potential

Dae-Yup Song

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1

Aug 4, 201322 citations

Alternative perspective on quantum tunneling and instantons

Frédérik Paradis, H. Kröger, G. Melkonyan +1 more

We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.

arXiv:quant-ph/0408051Physical Review AFeb 16, 20051 citation

Tunneling in asymmetric double well: instanton calculus

Alejandro Rivero

The level splitting formula of an asymmetric double well potential is calculated taking into account the multi-instanton contributions (dilute gas approximation). Results can be related with known semiclassical ones obtained with a truncated hamiltonian, and the symmetric case is easily recovered.

arXiv:quant-ph/0209072Sep 11, 20020 citations

Double well ground state energy splitting (or instanton flipping rate); rendering the implicit explicit

J H Hannay

A prime example of quantum tunnelling is the semiclassical 'energy splitting' of the levels of a symmetrical double well potential, or equivalently the flipping rate of an instanton. Curiously the accepted expression for the ground state energy splitting in terms of the (smooth) potential function has not been pursued to the full explicitness available from classical mechanics. This implicitness is rectified here.

arXiv:2403.18050Mar 26, 20240 citations

Analysis of nonperturbative fluctuations in a triple-well potential

JAVIER CASAHORRAN

We consider the quantum tunneling phenomenon in a well-behaved triple-well potential. As required by the semiclassical approximation we take into account the quadratic fluctuations over the instanton which represents as usual the localised finite-action solution of the euclidean equation of motion. The determinants of the quadratic differential operators at issue are evaluated by means of the Gelfang-Yaglom method. In doing so the explicit computation of the conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the individual frequencies derived from the non-equivalent minima of the potential. Eventually the physical effects of the multi-instanton configurations are included in this approach. As a matter of fact we obtain information about the energies of the ground-state and the two first excited levels of the discrete spectrum at issue.

Jan 1, 20013 citations